Skip to main content

Hyperbola Hidden within a Hyperboloid

A quadric surface is a surface with the equation \(Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0\). Depending on which coefficients are negative, positive, and zero, you can get ellipses, paraboloids, cylinders, cones, and hyperboloids out of this equation. What is interesting about quadric surfaces is that if you take a cross section of the surface, you can get a conic section (ellipse, parabola, or hyperbola) depending on how it's cut. The equation of the cross section is \(Ax^2+By^2+Cxy+Dx+Ey+F=0\). The way you find the equation of the cross section is a bit complicated. First, the equation of the quadric surface has to be simplified. If there is any \(G, H\), or \(I\), you must complete the square with the corresponding \(A, B,\) or \(C\). For any \(D, E\), or \(F\), you must replace \(x,y\),and \(z\) with the parameterized versions and use a \(\theta\) or \(\phi\) that will cancel out the \(xy, xz,\) and \(yz\). After that, put the equation of the plane in the form \(Ax + By + Cz = D\) and solve for one of the variables. Then, replace the variable in the simplified quadric surface equation and now you have the equation of a conic section. If there is any \(C, D\), or \(E\) in the conic section equation, do the same thing as before to get rid of them. After that, you have a recognizable formula for the cross section of the quadric surface.
 
The quadric surface I chose was a hyperboloid of one sheet with the equation \(\frac{x^2}{2^2} + \frac{y^2}{1^2}-\frac{z^2}{.5^2}=1\). This hyperboloid is twice as long in the \(x\) direction and half as tall in the \(z\) direction compared to a standard hyperboloid of one sheet. The intersecting plane is \(x+y+z=0\) which is a plane perpendicular to the vector [1,1,1]. The way my plane intersects the hyperboloid creates a hyperbola. Solving for \(z\) in the plane equation gets me \(z=-x-y\). Plugging that into the hyperboloid equation gets me \(\frac{x^2}{2^2} + \frac{y^2}{1^2}-\frac{(-x-y)^2}{.5^2}=1\). Simplifying gets \(\frac{-15x^2}{4} - 3y^2-8xy=1\). There is no \(D\) or \(E\) in this equation, so we just have to replace \(x\) and \(y\) with their parameterizations which are \(x=x\cos\theta-y\sin\theta\) and \(y=x\sin\theta+y\cos\theta\). The \(\theta\) needed for this simplification is \(\frac{1}{2}*\cot{\frac{-8}{-3.75+3}}\). I have no clue how to represent that in a better way, so from now on, we're gonna use decimal values rounded to four places. After plugging \(\theta\) in and simplifying, the equation we end up with is \(-7.3925x^2+.6425y^2=1\). This is the equation of a thin hyperbola with the vertices close to each other and on the y axis.

  The reason I chose this quadric surface and intersection is because I think that hyperbolas are the least utilized conic section. Sure, circles, ellipses, and parabolas have more real world practicality, but hyperbolas can be used to locate a point given its distance to three other points. A circle can't do that. As for the quadric surface, a hyperboloid is just a hyperbola rotated about its axis. So now, I get to demonstrate how to find the cross section of a quadric surface using not only a hyperbola, but also what is basically a hyperbola in 3D. I chose the coefficients for the hyperboloid and the intersecting plane to make finding the conic section a little more challenging. A standard hyperboloid and a level plane would be able to demonstrate how conic sections are found, but adding coefficients brings in a few more obstacles to tackle.

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more